Deformation and Vibration of an Oblique Elliptic Torus

The formulation used by the most of studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of the torus, application of those formulations has encountered great difficulty. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of the torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency.